Pre-Algebra Examples

$\frac{6 x^{3} + 14 x^{2} + 20 x + 12}{3 x + 4}$

Step 1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$3 x$

$4$

$6 x^{3}$

$14 x^{2}$

$20 x$

$12$

Step 2

Divide the highest order term in the dividend $6 x^{3}$ by the highest order term in divisor $3 x$ .

					$2 x^{2}$
	$3 x$	+	$4$		$6 x^{3}$	+	$14 x^{2}$	+	$20 x$	+	$12$

Step 3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$3 x$	+	$4$		$6 x^{3}$	+	$14 x^{2}$	+	$20 x$	+	$12$
			+	$6 x^{3}$	+	$8 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{3} + 8 x^{2}$

				$2 x^{2}$
$3 x$	+	$4$		$6 x^{3}$	+	$14 x^{2}$	+	$20 x$	+	$12$
			-	$6 x^{3}$	-	$8 x^{2}$

Step 5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$2 x^{2}$
$3 x$	+	$4$		$6 x^{3}$	+	$14 x^{2}$	+	$20 x$	+	$12$
			-	$6 x^{3}$	-	$8 x^{2}$
					+	$6 x^{2}$

Step 6

Pull the next terms from the original dividend down into the current dividend.

				$2 x^{2}$
$3 x$	+	$4$		$6 x^{3}$	+	$14 x^{2}$	+	$20 x$	+	$12$
			-	$6 x^{3}$	-	$8 x^{2}$
					+	$6 x^{2}$	+	$20 x$

Step 7

Divide the highest order term in the dividend $6 x^{2}$ by the highest order term in divisor $3 x$ .

				$2 x^{2}$	+	$2 x$
$3 x$	+	$4$		$6 x^{3}$	+	$14 x^{2}$	+	$20 x$	+	$12$
			-	$6 x^{3}$	-	$8 x^{2}$
					+	$6 x^{2}$	+	$20 x$

Step 8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$2 x$
$3 x$	+	$4$		$6 x^{3}$	+	$14 x^{2}$	+	$20 x$	+	$12$
			-	$6 x^{3}$	-	$8 x^{2}$
					+	$6 x^{2}$	+	$20 x$
					+	$6 x^{2}$	+	$8 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{2} + 8 x$

				$2 x^{2}$	+	$2 x$
$3 x$	+	$4$		$6 x^{3}$	+	$14 x^{2}$	+	$20 x$	+	$12$
			-	$6 x^{3}$	-	$8 x^{2}$
					+	$6 x^{2}$	+	$20 x$
					-	$6 x^{2}$	-	$8 x$

Step 10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$2 x^{2}$	+	$2 x$
$3 x$	+	$4$		$6 x^{3}$	+	$14 x^{2}$	+	$20 x$	+	$12$
			-	$6 x^{3}$	-	$8 x^{2}$
					+	$6 x^{2}$	+	$20 x$
					-	$6 x^{2}$	-	$8 x$
							+	$12 x$

Step 11

Pull the next terms from the original dividend down into the current dividend.

				$2 x^{2}$	+	$2 x$
$3 x$	+	$4$		$6 x^{3}$	+	$14 x^{2}$	+	$20 x$	+	$12$
			-	$6 x^{3}$	-	$8 x^{2}$
					+	$6 x^{2}$	+	$20 x$
					-	$6 x^{2}$	-	$8 x$
							+	$12 x$	+	$12$

Step 12

Divide the highest order term in the dividend $12 x$ by the highest order term in divisor $3 x$ .

				$2 x^{2}$	+	$2 x$	+	$4$
$3 x$	+	$4$		$6 x^{3}$	+	$14 x^{2}$	+	$20 x$	+	$12$
			-	$6 x^{3}$	-	$8 x^{2}$
					+	$6 x^{2}$	+	$20 x$
					-	$6 x^{2}$	-	$8 x$
							+	$12 x$	+	$12$

Step 13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$2 x$	+	$4$
$3 x$	+	$4$		$6 x^{3}$	+	$14 x^{2}$	+	$20 x$	+	$12$
			-	$6 x^{3}$	-	$8 x^{2}$
					+	$6 x^{2}$	+	$20 x$
					-	$6 x^{2}$	-	$8 x$
							+	$12 x$	+	$12$
							+	$12 x$	+	$16$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $12 x + 16$

				$2 x^{2}$	+	$2 x$	+	$4$
$3 x$	+	$4$		$6 x^{3}$	+	$14 x^{2}$	+	$20 x$	+	$12$
			-	$6 x^{3}$	-	$8 x^{2}$
					+	$6 x^{2}$	+	$20 x$
					-	$6 x^{2}$	-	$8 x$
							+	$12 x$	+	$12$
							-	$12 x$	-	$16$

Step 15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$2 x^{2}$	+	$2 x$	+	$4$
$3 x$	+	$4$		$6 x^{3}$	+	$14 x^{2}$	+	$20 x$	+	$12$
			-	$6 x^{3}$	-	$8 x^{2}$
					+	$6 x^{2}$	+	$20 x$
					-	$6 x^{2}$	-	$8 x$
							+	$12 x$	+	$12$
							-	$12 x$	-	$16$
									-	$4$

Step 16

The final answer is the quotient plus the remainder over the divisor.

$2 x^{2} + 2 x + 4 - \frac{4}{3 x + 4}$